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In the absence of free electric charge and electric current, the source-free Maxwell equations in a linear medium aretogether with the constitutive relations $\mathbf{D}=\epsilon \mathbf{E}$ and $\mathbf{B}=\mu \mathbf{H}$.

The Clausius gas equation can be writtenComparison with the virial expansion givesFor a hard-sphere gas of atomic radius $r_0$, the pair potential is infinite when the center separation is below $2r_0$ and zero otherwise. ThusThis is four times the volume of one atom and is the per-particle excluded volume appearing in the dilute equation of state.

Let the pair potential be $u(r)$, so that in the dilute limit only one interacting pair need be retained at a time. The leading Mayer f-function expansion isFor each pair, translation of its center coordinate supplies $V$ and the other $N-2$ particles supply $V^{N-2}$. Neglecting boundary corrections,The Taylor expansion $\log (1+x)=x+O(x^2)$ and $N(N-1)\simeq N^2$ in the thermodynamic limit giveUsing $P=-(\partial F/\partial V{)}_{T,N}$ and the ideal gas free energy,With number density $n=N/V$, this is $P/(nT)=1+B(T)n+O(n^2)$, so $B(T)$ is precisely the second virial coefficient.

Set $k_B=1$. After the Gaussian integral over momenta, the canonical partition function factors aswhereas the ideal gas has $Z_{\mathrm{ideal}}=Z_{\mathrm{kin}}{V}^N/N!$. HenceUsing the Helmholtz free energy $F=-T\log Z$ therefore gives

For a particle of mass $m$ and charge $q$, the Hamiltonian of a charged particle in an electromagnetic field isThe canonical momentum is $\mathbf{p}=m\dot{\mathbf{x}}+q\mathbf{A}$. Hamilton's equations are equivalent to the Lorentz force law

Multiplying the two Liouville–Green forms by $e^{x/2}$ givesThe first expression is in fact exact:and direct substitution gives

Putfor sufficiently large $x$. The Liouville–Green approximation givesSincewe obtain, after absorbing nonzero constants into the two solutions,

Use the standard elimination of the first derivative by settingThenso the transformed equation is